相关论文: Higher-dimensional Generalisations of the Euler To…
In low dimensional topology, we have some invariants defined by using solutions of some nonlinear elliptic operators. The invariants could be understood as Euler class or degree in the ordinary cohomology, in infinite dimensional setting.…
We consider a new variant of cosmological perturbation theory that has been designed specifically to include non-linear density contrasts on scales 100 Mpc, while still allowing for linear fluctuations on larger scales. This theory is used…
Direct links between generalized harmonic numbers, linear Euler sums and Tornheim double series are established in a more perspicuous manner than is found in existing literature. We show that every linear Euler sum can be decomposed into a…
Recently, the first author together with Jens Marklof studied generalizations of the classical three distance theorem to higher dimensional toral rotations, giving upper bounds in all dimensions for the corresponding numbers of distances…
A non-isospectral linear problem for an integrable 2+1 generalization of the non linear Schr\"odinger equation, which includes dispersive terms of third and fourth order, is presented. The classical symmetries of the Lax pair and the…
We study bounded ancient solutions of the Navier-Stokes equations. These are the solutions which are defined for all past time. In two space dimensions we prove that such solutions are either constant or functions of time only, depending on…
In this paper, for compressible Euler equations in multiple space dimensions, we prove the break-down of classical solutions with a large class of initial data by tracking the propagation of radially symmetric expanding wave including…
The Navier--Stokes (NS) equations describe fluid dynamics through a high-dimensional, nonlinear system of partial differential equations (PDEs). Despite their fundamental importance, their behavior in turbulent regimes remains incompletely…
The explicit formulation of the general inverse problem on conservation laws is presented for the first time. In this problem one aims to derive the general form of systems of differential equations that admit a prescribed set of…
Building on the work of Crouseilles and Faou on the 2D case, we construct $C^\infty$ quasi-periodic solutions to the incompressible Euler equations with periodic boundary conditions in dimension 3 and in any even dimension. These solutions…
We consider an integrable three-dimensional system of ordinary differential equations introduced by S.V. Kovalevskaya in a letter to G. Mittag-Leffler. We prove its isomorphism with the three-dimensional Euler top, and propose two…
In the language of differential geometry, the incompressible inviscid Euler equations can be written in vorticity-vector potential form as \begin{align*} \partial_t \omega + {\mathcal L}_u \omega &= 0\\ u &= \delta \tilde \eta^{-1}…
The Euler-$\alpha$ equations model the averaged motion of an ideal incompressible fluid when filtering over spatial scales smaller than $\alpha$. We show that there exists $\beta>1$ such that weak solutions to the two and three dimensional…
In this paper, the singularity formation of classical solutions for the compressible Euler equations with general pressure law is considered. The gradient blow-up of classical solutions is shown without any smallness assumption by the…
Whether the 3D incompressible Euler equations can develop a singularity in finite time from smooth initial data is one of the most challenging problems in mathematical fluid dynamics. This work attempts to provide an affirmative answer to…
We consider a complexification of the Euler equations introduced by \v{S}ver\'ak which conserves energy. We prove that these complex Euler equations are nonlinearly ill-posed below analytic regularity and, moreover, we exhibit solutions…
For a $C^1_{t,x}$ solution $u$ to the incompressible 3D Euler equations, the helicity $H(u(t))=\int_{\mathbb{T}^3} u \cdot \textrm{curl}\, u$ is constant in time. For general low-regularity weak solutions, it is not always clear how to…
We study blow-up rates and the blow-up profiles of possible asymptotically self-similar singularities of the 3D Euler equations, where the sense of convergence and self-similarity are considered in various sense. We extend much further, in…
The purpose of this article is to compute the normal form of a class of general quadratic Hamiltonian systems that generalizes naturally Euler's equations from the free rigid body dynamics.
A generalisation of the odd Bernoulli polynomials related to the quantum Euler top is introduced and investigated. This is applied to compute the coefficients of the spectral polynomials for the classical Lam\'e operator.